Solving Puzzles and Finding Numbers: The Mystery of 84

In today’s world, puzzles and numbers can often unlock hidden treasures of logical reasoning and mathematical beauty. One such intriguing puzzle revolves around the two-digit number 84. Let’s explore the journey of solving this puzzle and understanding the beauty of numbers.

The Riddle of 84

The answer to the riddle is 84. This two-digit number, when examined closely, hides several interesting properties. It is divisible by 3 and 4, which means it follows certain divisibility rules. Furthermore, the tens digit (8) is twice the ones digit (4).

Breaking Down the Clues

Let’s break down the clues to find our mystery number. The first clue tells us that 84 is a two-digit number, which means it ranges between 10 and 99. The second clue specifies that the number is divisible by 3 and 4. Since the least common multiple of 3 and 4 is 12, any number divisible by both 3 and 4 must also be divisible by 12.

The third clue reveals that the tens digit is twice the ones digit. Let’s denote the tens digit as T and the ones digit as O. The relationship is given by T 2O.

Considering the possibilities, since T must be a single digit (0-9), the valid values for O are as follows:

If O 1, then T 2, making the number 21. If O 2, then T 4, making the number 42. If O 3, then T 6, making the number 63. If O 4, then T 8, making the number 84.

Now, let’s check which of these numbers is divisible by 12:

21 is not divisible by 12. 42 is not divisible by 12. 63 is not divisible by 12. 84 is divisible by 12.

Hence, the two-digit number that meets all the criteria is 84.

Further Puzzles and Clues

Let’s explore another riddle that involves the same number: ‘You are also divisible by 2 and 6. When you add together one and two, you get three. You are twelve.’ This adds another layer of complexity. Clearly, the number 12 fits this description, but we already know that 84 is the answer to the first riddle. Let’s see how it fits:

To satisfy the clue, the units digit must be 1, 2, 3, or 4 because doubling 5 and over will not give you a single digit for the tens place. Additionally, the number must be divisible by 4 and be even. The closest candidates are 42 and 84.

Checking the divisibility rule, 42 is not divisible by 4 while 84 is. Therefore, we conclude that the number is 84.

Conclusion

The number 84 stands as a testament to the beauty and complexity of numbers, where simple riddles can unravel a series of mathematical properties. By breaking down the clues and applying divisibility rules, we can solve these puzzles and find hidden treasures in numbers. This exercise not only enhances logical reasoning skills but also deepens our understanding of mathematical principles.